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(1)

Twisted Alexander polynomials - an overview

Stefan Friedl

September 2010

(2)

Questions about knots

By a knotK we mean a closed embedded curve inS3.

We list some goals in knot theory. (2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(3)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(4)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(5)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface).

(2) Determine the genus g(K) of K. (2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(6)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface). Thegenus of K is the minimal genus among all Seifert surfaces.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(7)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface). Thegenus of K is the minimal genus among all Seifert surfaces.

Goal: determine the genusg(K) of a given knot

(2) Determine the genusg(K) of K.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(8)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(9)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ1(S3\Σ) is free.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(10)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ1(S3\Σ) is free. The minimal genus of such a Seifert surface is thefree genus ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(11)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ1(S3\Σ) is free. The minimal genus of such a Seifert surface is thefree genus ofK.

Goal: determine the free genusgfree(K) of a knot.

(2’) Determine the free genus of a given knot

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(12)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(13)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS3\K →S1

(3) Determine whether a given knot is fibered

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(14)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS3\K →S1 (i.e. a map such that the preimage of an interval is a surface times an interval).

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(15)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS3\K →S1 (i.e. a map such that the preimage of an interval is a surface times an interval). Note that a fiber is a genus minimizing Seifert surface.

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(16)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS3\K →S1 (i.e. a map such that the preimage of an interval is a surface times an interval). Note that a fiber is a genus minimizing Seifert surface.

Goal: determine whether a knotK is fibered.

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(17)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(18)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) A knot issliceif it bounds a smooth disk in D4.

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(19)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) A knot issliceif it bounds a smooth disk in D4. Goal: determine which knots are slice.

(4) Determine whether K is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(20)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(21)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of order n

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(22)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of ordernif there exists a homeomorphism ofS3 of order r

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

(23)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of ordernif there exists a homeomorphism ofS3 of order r which fixes an unknot pointwise andK setwise.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(24)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of ordernif there exists a homeomorphism ofS3 of order r which fixes an unknot pointwise andK setwise.

Goal: Determine which knots are periodic.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(25)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(26)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K its mirror image

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

(27)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K itsmirror imagei.e. the result of reflectingK in a plane.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

(28)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K itsmirror imagei.e. the result of reflectingK in a plane. A knot which equals its mirror image is calledamphichiral.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(29)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K itsmirror imagei.e. the result of reflectingK in a plane. A knot which equals its mirror image is calledamphichiral. Goal: Determine which knots are amphichiral.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

(30)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

(31)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism π1(S3\K1)→π1(S3\K2).

(7) Determine the partial order≥ of knots.

(32)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism π1(S3\K1)→π1(S3\K2).

This defines a partial order on the set of knots.

(7) Determine the partial order≥of knots.

(33)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism π1(S3\K1)→π1(S3\K2).

This defines a partial order on the set of knots. Goal: determine the partial order of knots.

(7) Determine the partial order≥of knots.

(34)

Questions about knots

By a knotK we mean a closed embedded curve inS3. We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

(35)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K.

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The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality

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The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti.

(38)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti. The infinite cyclic grouphti acts on H1( ˜X),

(39)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti. The infinite cyclic grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t±1].

(40)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti. The infinite cyclic grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t±1].

We write

H1(X;Z[t±1]) =H1( ˜X).

(41)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti. The infinite cyclic grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t±1].

We write

H1(X;Z[t±1]) =H1( ˜X).

We have a resolution

Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0 and we define

(42)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We haveH1(S3\K) =Zby Alexander duality and we denote by ˜X the infinite cyclic cover of X corresponding toπ1(X)→H1(X)→Z=hti. The infinite cyclic grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t±1].

We write

H1(X;Z[t±1]) =H1( ˜X).

We have a resolution

Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0 and we define

K(t) = det(D)∈Z[t±1].

(43)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We have a resolution Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0

and we define

K(t) = det(D)∈Z[t±1].

(1) IfAis a Seifert matrix, then D =At−At and hence

K(t) = det(At −At).

(44)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We have a resolution Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0

and we define

K(t) = det(D)∈Z[t±1].

(1) IfAis a Seifert matrix, then D =At−At and hence

K(t) = det(At −At).

This approach is very effective for knots but does not generalize well to 3-manifolds.

(45)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We have a resolution Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0

and we define

K(t) = det(D)∈Z[t±1].

(1) IfAis a Seifert matrix, then D =At−At and hence

K(t) = det(At −At).

(2) ∆K(t) can be computed easily using Fox calculus.

(46)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We have a resolution Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0

and we define

K(t) = det(D)∈Z[t±1].

(1) IfAis a Seifert matrix, then D =At−At and hence

K(t) = det(At −At).

(2) ∆K(t) can be computed easily using Fox calculus.

(3) ∆K(t) can also be expressed using Reidemeister-Milnor-Turaev torsion

(47)

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S3\K. We have a resolution Z[t±1]n D−→Z[t±1]n→H1(X,Z[t±1])→0

and we define

K(t) = det(D)∈Z[t±1].

(1) IfAis a Seifert matrix, then D =At−At and hence

K(t) = det(At −At).

(2) ∆K(t) can be computed easily using Fox calculus.

(3) ∆K(t) can also be expressed using Reidemeister-Milnor-Turaev torsion (which is my favorite view point!)

(48)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(49)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1]

(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(50)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and is well-defined up to multiplication by±tk.

(2) The Alexander polynomial of the trivial knot equals 1. (3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(51)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(52)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

The Alexander polynomial of the trefoil knot equalst−1−1 +t.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(53)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1

(4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(54)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 so the Alexander polynomial is not a complete invariant of knots

(4) The Alexander polynomial is unchanged under mutation. (5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(55)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(56)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1)

(6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(57)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1)

(this is a consequence of Poincar´e duality)

(6) ∆K(1) =±1 (7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(58)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(59)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(ForK a null-homologous knot in a homology sphere Σ we have

K(1) =|H1(Σ)|)

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(60)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(61)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(This is a consequence of ∆K(t) = det(At−At) whereAcan be a Seifert matrix of size 2g(K)×2g(K)).

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(62)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K)

and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(63)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(64)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆K(t) is monic.

(IfK is fibered andA a Seifert matrix for a fiber, then det(A) = 1,

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for somef(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(65)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆K(t) is monic.

(IfK is fibered andA a Seifert matrix for a fiber, then det(A) = 1, so the claim follows from ∆K(t) = det(At−At)).

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(66)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1]

(10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(67)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (IfD⊂D4 is a slice disk, this follows from Poincar´e duality applied to the pair (D4\D,S3\K))

(10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(68)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(69)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

i.e. the Alexander polynomial does not distinguish between mirror images

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(70)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(71)

Properties of the Alexander polynomial

LetK be a knot and ∆K(t) its Alexander polynomial.

(1) ∆K(t)∈Z[t±1] and well-def. up to multiplication by±tk. (2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t−1) (6) ∆K(1) =±1

(7) deg(∆K(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t−1) for some f(t)∈Z[t±1] (10) ∆K(t) = ∆K(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

(72)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD

Denote the epimorphism π →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1]

(73)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD (e.g. Zor C)

Denote the epimorphism π→Zbyφ and the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1]

(74)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφ

and the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1]

(75)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X.

Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1]

(76)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations

andZ[π] acts onR[t±1]⊗Rn=Rn[t±1]

(77)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

(78)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

(79)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

Consider Cα(X;Rn[t±1]) :=C( ˜X)⊗Z[π]Rn[t±1]

(80)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

Consider Cα(X;Rn[t±1]) :=C( ˜X)⊗Z[π]Rn[t±1] (this is a chain complex over the ringR[t±1])

(81)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

Consider Cα(X;Rn[t±1]) :=C( ˜X)⊗Z[π]Rn[t±1]

(this is a chain complex over the ringR[t±1]) and its homology Hα(X;Rn[t±1]).

(82)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

Consider Cα(X;Rn[t±1]) :=C( ˜X)⊗Z[π]Rn[t±1]

(this is a chain complex over the ringR[t±1]) and its homology Hα(X;Rn[t±1]). Pick a resolution

Rn[t±1]k D−→Rn[t±1]l →Hα(X;Rn[t±1])→0

(83)

Twisted Alexander polynomials: homological definition

LetK ⊂S3 and α:π=π1(S3\K)→GL(n,R) a representation over a UFD Denote the epimorphismπ →Zbyφand the universal cover ofX =S3\K by ˜X. Z[π] acts onC( ˜X) by deck transformations andZ[π] acts onR[t±1]⊗Rn=Rn[t±1] as follows:

g·(p(t)⊗v) =tφ(g)p(t)⊗α(g)v.

Consider Cα(X;Rn[t±1]) :=C( ˜X)⊗Z[π]Rn[t±1]

(this is a chain complex over the ringR[t±1]) and its homology Hα(X;Rn[t±1]). Pick a resolution

Rn[t±1]k D−→Rn[t±1]l →Hα(X;Rn[t±1

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Because every doubly primitive knot is shown to be fibered in [9], and then, it is well-known that the genus of a fibered knot is determined by the Alexander polynomial.. Also, by

For any knot with the Alexander polynomial of degree 2, we can prove the problem for $GL(2,\mathbb{Z}/p)$ -representations. There exists a solution of $\triangle_{K}(t)\equiv 0$

the second coefficient of the twisted Alexander polynomial $\Delta_{K,\rho}(t)$ associated to parabolic.. representations $\rho$ (not only for the holonomy representation

speaking, the Riley polynomial gives a defining equation of the nonabelian part of the space of conjugacy classes of $SL(2, \mathbb{C})$ -representations of a 2-bridge knot..